Pub Date : 2023-12-29DOI: 10.1134/s00122661230110071
V. I. Maksimov
Abstract
We consider a guaranteed control problem for a nonlinear distributed equation of diffusion type. The problem is essentially to construct a feedback control algorithm ensuring that the solution of a given equation tracks the solution of a similar equation subjected to an unknown disturbance. The case in which a discontinuous unbounded function can be a feasible disturbance is studied. We solve the problem under conditions of inaccurate measurement of solutions of each of the equations at discrete instants of time and indicate a solution algorithm robust under information noise and calculation errors.
{"title":"On a Positional Control Problem for a Nonlinear Equation with Distributed Parameters","authors":"V. I. Maksimov","doi":"10.1134/s00122661230110071","DOIUrl":"https://doi.org/10.1134/s00122661230110071","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We consider a guaranteed control problem for a nonlinear distributed equation of diffusion\u0000type. The problem is essentially to construct a feedback control algorithm ensuring that the\u0000solution of a given equation tracks the solution of a similar equation subjected to an unknown\u0000disturbance. The case in which a discontinuous unbounded function can be a feasible disturbance\u0000is studied. We solve the problem under conditions of inaccurate measurement of solutions of each\u0000of the equations at discrete instants of time and indicate a solution algorithm robust under\u0000information noise and calculation errors.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139072371","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-29DOI: 10.1134/s00122661230110095
A. S. Fursov, P. A. Krylov
Abstract
The problem of constructing the graph of states of a switched affine system closed by a static state feedback is considered. To solve this problem, a constructive algorithm based on the study of the consistency of systems of linear algebraic inequalities is proposed.
{"title":"On the Construction of the Graph of Discrete States of a Switched Affine System","authors":"A. S. Fursov, P. A. Krylov","doi":"10.1134/s00122661230110095","DOIUrl":"https://doi.org/10.1134/s00122661230110095","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> The problem of constructing the graph of states of a switched affine system closed by a\u0000static state feedback is considered. To solve this problem, a constructive algorithm based on the\u0000study of the consistency of systems of linear algebraic inequalities is proposed.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139066391","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-29DOI: 10.1134/s0012266123011006x
M. I. Gomoyunov
Abstract
We consider the optimal control problem of minimizing the terminal cost functional for a dynamical system whose motion is described by a differential equation with Caputo fractional derivative. The relationship between the necessary optimality condition in the form of Pontryagin’s maximum principle and the Hamilton–Jacobi–Bellman equation with so-called fractional coinvariant derivatives is studied. It is proved that the costate variable in the Pontryagin maximum principle coincides, up to sign, with the fractional coinvariant gradient of the optimal result functional calculated along the optimal motion.
{"title":"On the Relationship Between the Pontryagin Maximum Principle and the Hamilton–Jacobi–Bellman Equation in Optimal Control Problems for Fractional-Order Systems","authors":"M. I. Gomoyunov","doi":"10.1134/s0012266123011006x","DOIUrl":"https://doi.org/10.1134/s0012266123011006x","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We consider the optimal control problem of minimizing the terminal cost functional for a\u0000dynamical system whose motion is described by a differential equation with Caputo fractional\u0000derivative. The relationship between the necessary optimality condition in the form of\u0000Pontryagin’s maximum principle and the Hamilton–Jacobi–Bellman equation with so-called\u0000fractional coinvariant derivatives is studied. It is proved that the costate variable in the\u0000Pontryagin maximum principle coincides, up to sign, with the fractional coinvariant gradient of\u0000the optimal result functional calculated along the optimal motion.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139066404","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-29DOI: 10.1134/s00122661230110125
A. P. Krishchenko, O. A. Podderegin
Abstract
We study a model of a predator–prey system with possible infection of prey in the form of a three-dimensional system of ordinary differential equations. Using the localization method of compact invariant sets, the existence of an attractor is proved and a compact positively invariant set is found that estimates its position. The conditions for the extinction of populations and the existence of equilibria are found. A numerical method for finding a Hopf bifurcation of the inner equilibrium is proposed and an example of an arising stable limit cycle is given.
{"title":"Hopf Bifurcation in a Predator–Prey System with Infection","authors":"A. P. Krishchenko, O. A. Podderegin","doi":"10.1134/s00122661230110125","DOIUrl":"https://doi.org/10.1134/s00122661230110125","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We study a model of a predator–prey system with possible infection of prey in the form of\u0000a three-dimensional system of ordinary differential equations. Using the localization method of\u0000compact invariant sets, the existence of an attractor is proved and a compact positively invariant\u0000set is found that estimates its position. The conditions for the extinction of populations and the\u0000existence of equilibria are found. A numerical method for finding a Hopf bifurcation of the inner\u0000equilibrium is proposed and an example of an arising stable limit cycle is given.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139066664","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-29DOI: 10.1134/s00122661230110010
A. V. Arutyunov, Z. T. Zhukovskaya, S. E. Zhukovskiy
Abstract
We consider autonomous differential inclusions with nonlinear boundary conditions. Sufficient conditions for the existence of solutions in the class of absolutely continuous functions are obtained for these inclusions. It is shown that the corresponding existence theorem applies to the Cauchy problem and the antiperiodic boundary value problem. The result is used to derive a new mean value inequality for continuously differentiable functions.
{"title":"On Nonlinear Boundary Value Problems for Differential Inclusions","authors":"A. V. Arutyunov, Z. T. Zhukovskaya, S. E. Zhukovskiy","doi":"10.1134/s00122661230110010","DOIUrl":"https://doi.org/10.1134/s00122661230110010","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We consider autonomous differential inclusions with nonlinear boundary conditions.\u0000Sufficient conditions for the existence of solutions in the class of absolutely continuous functions\u0000are obtained for these inclusions. It is shown that the corresponding existence theorem applies to\u0000the Cauchy problem and the antiperiodic boundary value problem. The result is used to derive a\u0000new mean value inequality for continuously differentiable functions.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139066400","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-29DOI: 10.1134/s00122661230110034
V. S. Mokeichev, A. M. Sidorov
Abstract
We propose a new approach to the solvability of ordinary as well as partial differential equations in the theory of linear differential equations and also in the theory of integral equations.
摘要 我们提出了线性微分方程理论和积分方程理论中常微分方程和偏微分方程可解性的新方法。
{"title":"Solvability of Linear Differential Equations","authors":"V. S. Mokeichev, A. M. Sidorov","doi":"10.1134/s00122661230110034","DOIUrl":"https://doi.org/10.1134/s00122661230110034","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We propose a new approach to the solvability of ordinary as well as partial differential\u0000equations in the theory of linear differential equations and also in the theory of integral equations.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139066407","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-29DOI: 10.1134/s00122661230110101
A. V. Yurchenkov, I. R. Belov
Abstract
We consider a discrete-time-invariant system with multiplicative noise with implementation in the state space. The exogenous disturbance is chosen from the class of time-invariant ergodic sequences of nonzero colorness. We consider the level of mean anisotropy of the exogenous disturbance to be bounded by a known value. Conditions for the anisotropic norm to be bounded by a given number are obtained in terms of solving a matrix system of inequalities with a convex constraint of a special type. It is demonstrated how, on the basis of the obtained conditions, to construct a static state control that ensures the minimum value of the anisotropic norm of the system enclosed by this control.
{"title":"Bounded Real Lemma for the Anisotropic Norm of Time-invariant Systems with Multiplicative Noises","authors":"A. V. Yurchenkov, I. R. Belov","doi":"10.1134/s00122661230110101","DOIUrl":"https://doi.org/10.1134/s00122661230110101","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We consider a discrete-time-invariant system with multiplicative noise with\u0000implementation in the state space. The exogenous disturbance is chosen from the class of\u0000time-invariant ergodic sequences of nonzero colorness. We consider the level of mean anisotropy of\u0000the exogenous disturbance to be bounded by a known value. Conditions for the anisotropic norm\u0000to be bounded by a given number are obtained in terms of solving a matrix system of inequalities\u0000with a convex constraint of a special type. It is demonstrated how, on the basis of the obtained\u0000conditions, to construct a static state control that ensures the minimum value of the anisotropic\u0000norm of the system enclosed by this control.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139066673","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-29DOI: 10.1134/s00122661230110022
A. N. Kanatnikov, O. S. Tkacheva
Abstract
A model of interaction between the human immunodeficiency virus and the human immune system is considered. Equilibria in the state space of the system and their stability are analyzed, and the ultimate bounds of the trajectories are constructed. It has been proved that the local asymptotic stability of the equilibrium corresponding to the absence of disease is equivalent to its global asymptotic stability. The loss of stability is shown to be caused by a transcritical bifurcation.
{"title":"Behavior of Trajectories of a Four-Dimensional Model of HIV Infection","authors":"A. N. Kanatnikov, O. S. Tkacheva","doi":"10.1134/s00122661230110022","DOIUrl":"https://doi.org/10.1134/s00122661230110022","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> A model of interaction between the human immunodeficiency virus and the human\u0000immune system is considered. Equilibria in the state space of the system and their stability are\u0000analyzed, and the ultimate bounds of the trajectories are constructed. It has been proved that the\u0000local asymptotic stability of the equilibrium corresponding to the absence of disease is equivalent\u0000to its global asymptotic stability. The loss of stability is shown to be caused by a transcritical\u0000bifurcation.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139066389","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-15DOI: 10.1134/s00122661230130013
N. V. Zaitseva
Abstract
The paper considers nonlocal problems with integral conditions for hyperbolic equations with the Bessel differential operator whose statement substantially depends on the intervals where the parameter occurring in this operator varies. The well-posedness of these problems is studied according to a unified scheme based on the classical method of separation of variables, which is also used to study nonclassical problems with integral conditions for equations of elliptic–hyperbolic type containing the Bessel operator in one or two variables as well.
{"title":"Mixed Problems with Integral Conditions for Hyperbolic Equations with the Bessel Operator","authors":"N. V. Zaitseva","doi":"10.1134/s00122661230130013","DOIUrl":"https://doi.org/10.1134/s00122661230130013","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> The paper considers nonlocal problems with integral conditions for hyperbolic equations\u0000with the Bessel differential operator whose statement substantially depends on the intervals where\u0000the parameter occurring in this operator varies. The well-posedness of these problems is studied\u0000according to a unified scheme based on the classical method of separation of variables, which is\u0000also used to study nonclassical problems with integral conditions for equations of\u0000elliptic–hyperbolic type containing the Bessel operator in one or two variables as well.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138692327","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-01DOI: 10.1134/s0012266123120054
Abstract
Equations and systems of elliptic type with power-law nonlinearities are considered. Such equations are found in modeling distributed robotic formations, as well as in chemical kinetics, biology, astrophysics, and many other fields. The problem of constructing multidimensional exact solutions is studied. It is proposed to use a special type of ansatz that reduces the problem to solving systems of algebraic equations. A number of multiparameter families of new exact multidimensional solutions (both radially symmetric and anisotropic) represented by explicit formulas are obtained. Examples are given to illustrate the exact solutions found.
{"title":"On Exact Solutions of a Multidimensional System of Elliptic Equations with Power-Law Nonlinearities","authors":"","doi":"10.1134/s0012266123120054","DOIUrl":"https://doi.org/10.1134/s0012266123120054","url":null,"abstract":"<span> <h3>Abstract</h3> <p> Equations and systems of elliptic type with power-law nonlinearities are considered. Such equations are found in modeling distributed robotic formations, as well as in chemical kinetics, biology, astrophysics, and many other fields. The problem of constructing multidimensional exact solutions is studied. It is proposed to use a special type of ansatz that reduces the problem to solving systems of algebraic equations. A number of multiparameter families of new exact multidimensional solutions (both radially symmetric and anisotropic) represented by explicit formulas are obtained. Examples are given to illustrate the exact solutions found. </p> </span>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139979120","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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